Homogeneous Diophantine approximation inS-integers

Abstract

In this paper we generalize classical results in Diophantine approximation to the setting of an arhitrary numher field in the context of the ring of 5-integers. Specifically, we present theorems pertaining to simultaneous approximations of linear forms and develop the notion of badly approximable ^-systems. In addition, we expand the subject of the geometry of numbers over the adele ring of a number field by developing the concept of the adelic polar body. This theory is then used to produce transference theorems in this general situation.

Badly approximable S-numbers and absolute Schmidt games

“…To prove Proposition 2.1, we will make use of the following (special case of) lemma of Burger [Bur92]:…”

Simultaneous Approximation by Conjugate Algebraic Numbers in Fields of Transcendence Degree One

“…This follows from Lemma 3.1 (ii) and Theorem 3.7 of [2], in the case where E = E * = K m and where g is the usual bilinear form θ :…”

On Mahler's compound bodies

“…We begin by briefly of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870003202X recalling the adelic polar body as described in [3] and then considering the local situation.…”

“…Here the constants implied the Vinogradov symbol depend only upon the number field k and N, and are explicitly given in [3].…”

“…We state this theorem in Section 2. Recently in [3] we further expanded the subject by introducing the adelic polar body. We recall the basic results in Section 6.…”